In this paper, we present a bilevel optimal motion planning (BOMP) model for autonomous parking. The BOMP model treats motion planning as an optimal control problem, in which the upper level is designed for vehicle nonlinear dynamics, and the lower level is for geometry collision-free constraints. The significant feature of the BOMP model is that the lower level is a linear programming problem that serves as a constraint for the upper-level problem. That is, an optimal control problem contains an embedded optimization problem as constraints. Traditional optimal control methods cannot solve the BOMP problem directly. Therefore, the modified approximate Karush–Kuhn–Tucker theory is applied to generate a general nonlinear optimal control problem. Then the pseudospectral optimal control method solves the converted problem. Particularly, the lower level is the \(J_2\)-function that acts as a distance function between convex polyhedron objects. Polyhedrons can approximate objects in higher precision than spheres or ellipsoids. As a result, a fast high-precision BOMP algorithm for autonomous parking concerning dynamical feasibility and collision-free property is proposed. Simulation results and experiment on Turtlebot3 validate the BOMP model, and demonstrate that the computation speed increases almost two orders of magnitude compared with the area criterion based collision avoidance method.

中文翻译：

---

双层最优运动计划（BOMP）模型及其在自动泊车中的应用

在本文中，我们提出了一种用于自主停车的双层最优运动计划（BOMP）模型。BOMP模型将运动计划视为最佳控制问题，其中上层设计用于车辆非线性动力学，下层设计用于无几何碰撞约束。BOMP模型的重要特征是，下层是线性规划问题，是上层问题的约束。即，最优控制问题包含嵌入式优化问题作为约束。传统的最优控制方法不能直接解决BOMP问题。因此，将改进的近似Karush–Kuhn–Tucker理论应用于生成一般的非线性最优控制问题。然后，伪谱最优控制方法解决了转换后的问题。尤其，\（J_2 \） -充当凸多面体对象之间的距离函数的函数。多面体可以比球体或椭球体更高的精度逼近对象。因此，提出了一种具有动态可行性和无碰撞性的快速高精度BOMP自动停车算法。在Turtlebot3上的仿真结果和实验验证了BOMP模型，并证明与基于面积准则的防撞方法相比，计算速度提高了近两个数量级。